Divergence Theorem

The divergence theorem, sometimes known as the Gauss theorem or the Ostrogradsky theorem, is a fundamental statement in multivariable calculus. It relates the divergence of a vector field over a closed surface to the flux (or flux) of that field. This theorem is used in various fields such as physics, engineering, and computer graphics. Let's understand this theorem in detail, break it down into more accessible concepts and use examples to aid understanding.

Understanding divergence

Before we dive deep into the theorem, let’s take a moment and understand what “divergence” means in mathematics. The divergence of a vector field is a scalar function that measures the magnitude of the source or sink of the field at a given point, in simple terms, how much the vector field diverges from or converges towards that point.

Consider a three-dimensional vector field F = (F₁, F₂, F₃) The divergence of the vector field F is given by:

div( F ) = ∇ ⋅ F = ∂F₁/∂x + ∂F₂/∂y + ∂F₃/∂z

In this expression, the symbol ∇ (nabla) denotes the vector differential operator, and the dot product with F gives us the divergence.

Divergence theorem statement

The divergence theorem states:

Let F be a continuously differentiable vector field over a region V in ℝ³ with a piecewise smooth boundary S The divergence theorem relates the flux of the vector field through the closed surface S to the divergence of the vector field inside the volume V :

∫∫∫ V (∇ ⋅ F ) dV = ∫∫ S ( F ⋅ n ) dS

Where:

• dV is the differential volume element.
• dS is the differential surface element.
• n is the unit normal vector pointing outward on the surface.

Visual example

To understand this theorem, imagine a simple 3D sphere such as a cube. Here's what the relation represents:

Consider a cube in 3D space where the vector field F represents the flow of fluid or air through the cube. The "divergence" of F at a point within the cube tells us how much "fluid" is being added or removed at that point.

In this diagram, the blue arrows indicate the directions of the vector field within the cube. Divergence theory helps us measure the net "outflow" or "inflow" of field across all faces of the cube, by evaluating the divergence within and making sure it matches the flow emanating from the surface.

Algorithm and calculation example

Let us consider a practical example with simple calculations to illustrate the application of the divergence theorem:

Suppose we have a vector field F (x, y, z) = (x², y², z²) in a spherical volume V of radius R centered at the origin. The boundary of this volume S is the sphere itself.

We need to verify the divergence theorem by calculating both sides of the equation:

Step 1: Calculate the variance of F

From the definition of divergence, we calculate:

div( F ) = ∂/∂x(x²) + ∂/∂y(y²) + ∂/∂z(z²) = 2x + 2y + 2z

Step 2: Evaluate the volume integral

We carry out the calculation:

∫∫∫ V (2x + 2y + 2z) dV = ∫∫∫ V 2(x + y + z) dV

Because of symmetry, all terms relating to x, y and z will integrate to zero on the sphere because of symmetry around the origin.

Step 3: Evaluate the surface integral

To find the surface integral, we need the external normal unit vector n to the sphere. For a sphere of radius R, at any point on the surface, n = (x/R, y/R, z/R).

We carry out the calculation:

∫∫ S ( F ⋅ n ) dS = ∫∫ S ((x², y², z²) ⋅ (x/R, y/R, z/R)) dS

Adding this up, we get:

= ∫∫ S (x³/R + y³/R + z³/R) dS = 1/R ∫∫ S (x³ + y³ + z³) dS

By using spherical coordinates and integrating over a spherical surface, this integral becomes a constant that gives a measure of how the vector field behaves at the boundary.

Verification of the theorem

After calculating both integrals on each side of the equation for the Divergence theorem, you will find that under appropriate symmetric conditions, such as a sphere, both calculations give the same result. Therefore, the theorem is true.

Examples of calculations with geometric shapes

Example 1: Simple sector cube

Let us calculate this for a unit cube, which for simplicity is defined in the first octant, with vertices at (0,0,0) and (1,1,1), where the vector field F (x, y, z) = (x, y, z)

Volume integral

The deviation is as follows:

div( F ) = ∂/∂x(x) + ∂/∂y(y) + ∂/∂z(z) = 1 + 1 + 1 = 3

The volume integral is as follows:

∫∫∫ V 3 dV = 3 × Volume of cube = 3 × 1 = 3

Surface integrals

For each face of the cube, consider the orientation of the vector and calculate the flux over the faces and the contribution to the sum:

Top surface (z=1 face)

n = (0, 0, 1), F = (x, y, 1), F ⋅ n = 1 → Flux = ∫∫ x=0,1; y=0,1 1 dxdy = 1

Bottom surface (z=0 face)

n = (0, 0, -1), F = (x, y, 0), F ⋅ n = 0 → Flux = 0

Similarly, calculate the other arms and find the total flux = 3.

Both the volume and surface integrals return the same value, which satisfies the divergence theorem.

Example 2: Cylinder with radial vector field

Consider a vector field with circular symmetry for a cylindrical region x² + y² ≤ 1, 0 ≤ z ≤ h such that F (x, y, z) = (x, y, z²)

Volume integral

div( F ) = ∂/∂x(x) + ∂/∂y(y) + ∂/∂z(z²) = 1 + 1 + 2z ∫∫∫ V (2 + 2z) dV over transformed cylindrical coordinates.

Surface integrals

Investigate the flow through the side and top/bottom surfaces by applying external normals and integrating, confirming equivalence with the volume approach.

Conclusion

The divergence theorem beautifully connects the behavior of a field through a closed surface to its generalized source/expulsion in a volume. By translating seemingly complex 3D phenomena into the languages of surface and volume integrals, it facilitates understanding and analysis in physics, especially when assessing fluid continuity or electromagnetic fields.

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